PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 69 > pp. 287-304

A RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM

By Y. F. Gui and W.-B. Dou

Full Article PDF (322 KB)

Abstract:
There are some limitations on the statement of classic Helmholtz theorem although it has abroad application. Actually, it only applies to simply connected domain with single boundary surface and does not provide any conclusion about the domain where discontinuities of field function exist. However, discontinuity is often encountered in practice, for example, the location of surface sources or interface of two kinds of medium. Meanwhile, most existing versions of Helmholtz theorem are imprecise and imperfect to some extent. This paper not only tries to present a precise statement and rigorous proof on classic Helmholtz theorem with the accuracy of mathematical language and logical strictness, but also generalizes it to the case of multiply connected domain and obtains a generalized Helmholtz theorem in the sense of Lebesgue measure and Lebesgue integral defined on three-dimensional Euclidean space. Meanwhile, our proof and reasoning are more sufficient and perfect.

As an important application of the generalized Helmholtz theorem, the concepts of irrotational and solenoidal vector function are emphasized. The generalized Helmholtz theorem and the present conclusion should have important reference value in disciplines including vector analysis such as electromagnetics.

Citation:
Y. F. Gui and W.-B. Dou, "A Rigorous and Completed Statement on Helmholtz Theorem," Progress In Electromagnetics Research, Vol. 69, 287-304, 2007.
doi:10.2528/PIER06123101
http://www.jpier.org/PIER/pier.php?paper=06123101

References:
1. Jackson, J. D., Classical Electrodynamics, 2nd edition, Wiley, NY, 1975.

2. Stratton, J. A., Electromagnetic Theory, John Wiley & Sons, New York, 1941.

3. Sommerfeld, A., Electromagnetic Theory, Academic Press, New York, 1952.

4. Collin, R. E., Field Theory of Guided Waves, Mc Graw-Hill Book Co., New York, 1960.

5. Chew, W. C., Waves and Fields in Inhomogenous Media, Van Nostrand Teinhold, New York, 1990.

6. Cheng, D. K., Field and Wave Electromagnetics, Addision-Wesley Publishing Company. Inc., 1983.

7. Kong, J. A., Electromagnetic Wave Theory, 2nd edition, Wiley, New York, 1990.

8. Lin, W. G., Microwave Theory and Techniques, 671, 1979.

9. Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, Part II, McGraw-Hill Book Company, New York, 1953.

10. Courant, R., Methods of Mathematical Physics, Vol. 1, Vol. 1, Interscience Publisher, New York, 1953.

11. Landau, L. D. and E. M. Lifshitz, The Classical Theory of Fields, Pergamon Press, Oxford, 1962.

12. Song, W., Dyadic Green Function and Operator Theory of Electromagnetic Field, Science and technique University of China Press, 1991.

13. Rudin, W., Functional Analysis, McGraw-Hill, 1973.

14. Conway, J. B., A Course in Functional Analysis, Springer-Verlag, 1985.

15. Kreyszig, E., Introductory Functional Analysis with Application, John Wiley & Sons, 1978.

16. Tuma, J. J. and R. A.Walsh, Engineering Mathematics Handbook, 4th edition, McGraw-Hill Companies, Inc., 1998.

17. Zhou, X. L., "On independence, completeness of Maxwells equations and uniqueness theorems in electromagnetics," Progress In Electromagnetics Research, Vol. 64, 117-134, 2006.
doi:10.2528/PIER06061302

18. Zhou, X. L., "On uniqueness theorem of a vector function," Progress In Electromagnetics Research, Vol. 65, 93-102, 2006.
doi:10.2528/PIER06081202

19. Zhou, X. L., "On Helmhotlz's theorem and its interpretations," Accepted for publication in Journal of Electromagnetic Waves and Application, Vol. 21, No. 4, 471-483, 2007.


© Copyright 2014 EMW Publishing. All Rights Reserved